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Area of an Ellipse
Volume of a Prism

Engineering Aid 3 - Beginning Structural engineering guide book
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Irregular  Areas Irregular areas are those areas that do not fall within a definite standard shape. As you already have learned, there are formulas for computing the area of a circle, a rectangle, a triangle, and so on. However, we do not have a standard formula for computing the area of an irregular   shaped   plane,   unless   we   use   higher mathematics (calculus), and integrate incremental areas using   lower   and   upper   limits   that   define   the boundaries. As  an  EA,  however,  most  areas  you  will  be concerned  with  are  those  you  will  meet  in  plane surveying.  In  most  surveys,  the  computed  area  is  the horizontal projection of the area rather than the actual surface of the land. The fieldwork in finding areas consists of a series of angular and linear measurements, defining the outline of whatever the shape is of the area concerned,  and  forming  a  closed  traverse.  The following  office  computation  methods,  which  you  will learn as you advance in rate, are: 1.   Plotting   the   closed   traverse   to   scale   and measuring  the  enclosed  area  directly  with  a  polar planimeter  (used  only  where  approximate  results  are required, or for checking purposes). 2.  Subdividing  the  area  into  a  series  of  triangles, and taking the summation of all the areas of these triangles. 3. Computing the area using the coordinates of the individual points of the traverse (called coordinate method). 4. Computing the area by means of the balanced latitude  and  departure,  and  calculated  DOUBLE MERIDIAN DISTANCES of each course (called the DMD  method). 5. Computing the area by counting squares; this method is nothing but just superimposing small squares plotted on a transparent paper having the same scale as the plotted traverse (or of known graphical ratio) and counting the number of squares within the traverse. The smaller the squares, the closer to the approximate area you will get. 6. Computing an irregular area bounded by a curve and  perpendicular  lines,  as  shown  in  figure  1-16.  Here, you can use the TRAPEZOIDAL RULE. The figure is considered as being made up of a series of trapezoids, all of them having the same base and having common Figure 1-16.-Irregular area by trapezoidal rule. distances  between  offsets.  The  formula  in  computing the total area is as follows: For the present time, try to find the areas of irregular figures by subdividing the area to series of triangles and by the method of counting the squares. There are also areas of spherical surfaces and areas of portions of a sphere. For other figures not covered in this training manual, consult any text on plane and solid  geometry. DETERMINING  VOLUMES From  the  preceding  section  you  learned  the formulas  for  computing  the  areas  of  various  plane figures.  These  plane  areas  are  important  in  the computation of VOLUMES, as you will see later in this section. When  plane  figures  are  combined  to  form  a three-dimensional   object,   the   resulting   figure   is 1-14







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